## Abstract

On the basis of the linearized theory of water waves, the present study has demonstrated a semi-analytical method to assess the hydrodynamic performance of a pair of partially immersed barriers just above a thick bottom-standing barrier. By means of the eigenfunction expansion method, a system of the first kind Fredholm-type integral equation involving a horizontal component of velocity as unknown functions is developed for the interaction of water waves with both types of barriers. The multiterm Galerkin approximation is adopted to determine these unknown functions having square root singularities at the submerged edge of the thin barriers and one-third singularities at the corners of the thick barrier. In order to overcome such types of singularities, Chebychev polynomials for half-singularities and ultra-spherical Gegenbauer polynomials for one-third singularities with suitable weight functions have been taken into consideration. The numerical examples of both reflection and transmission coefficients are presented to examine the hydrodynamic performance of breakwater. Some fascinating results like resonant frequencies are obtained for practical engineering. At the same time, reflection coefficients for the present breakwater agree reasonably for the limiting cases with the previously available results.

## Introduction

In the past few decades, the vertical breakwater has paid considerable attention for its profound impact on the mitigation of the wave reflection. Vertical breakwaters are extensively used in ocean engineering to defend ports and operation areas for ship loading and unloading. These breakwaters also have provided a protection for the coastal regions from the intrusion of littoral drift. By utilizing the vertical barriers, an extremely stable structure has developed even in rough, deep seas and such cause leads to their use throughout the world.

Water wave scattering by vertical barriers becomes significant in the study of water wave problems from the last few decades. The scattering of water wave problems by appropriate arrangement of thin vertical barriers in finite depth water or deep water has been carried out by a large number of researchers, in the literature within the framework of the linear theory of water waves. In the context of vertical barriers, the early works of Ursell [1,2] motivated many researchers to deal with the problems of scattering and radiation of water waves by vertical barriers. Dean [3] exercised the complex variable method to encounter the scattering problem by the submerged plane barrier. Meanwhile, many scientists and researchers have devoted their efforts to improve different methods from time to time to reveal these problems in deep water, for example, Stoker [4], Levine and Rodemich [5], Lewin [6], and Mei [7]. Also, some numerical and semi-analytical methods have been applied to solve the problems in finite depth water such as Liu and Abbaspour [8], Lasoda et al. [9], Abul-Azm [10], and Porter and Evans [11].

In the offshore regions, the breakwaters in the form of thick barriers have increased over the past few decades. Water wave scattering by different structures of thick barriers has been investigated widely by the marine scientists and researchers. On the grounds of the huge sizes of these structures as compared to the wavelength of the ocean waves, the thickness of the configuration plays a major role in the analysis. The study of Mei and Black [12] is considered to be the pioneer work in water wave scattering by thick vertical barriers with a rectangular cross-section in uniform finite depth. With the assistance of the conformal mapping method, Guiney et al. [13] investigated the wave propagation problem by including the effect of thickness in a vertical wall of a rectangular cross-section. The problem of scattering of a normally incident surface wave train by an obstacle in the form of a thick vertical barrier of four different basic configurations was analyzed by Kanoria et al. [14] using the multiterm Galerkin technique involving ultra-spherical Gegenbauer polynomials. Later, Mandal and Kanoria [15] considered the oblique incident wave train to extend the work of Kanoria et al. [14]. Sturova [16] introduced an efficient tool for scrutinizing the behavior of long surface waves generated by periodic disturbances in a fluid with an uneven bottom of a particular type.

In the recent decades, analysis of both half and one-third singularities in the integral equations that arise in the problems of water waves is a matter of contention among the researchers. Ray et al. [17] studied wave propagation over submarine trench in the presence of a thin vertical partially immersed barrier for its three different positions. They discussed different types of integrable singularities (square root and cube root) at the end points for solving the integral equations. Ray et al. [18] also used simple polynomials as basis functions to tackle singularities in the problem of wave propagation over an infinite step in the presence of a thin vertical barrier of four different geometrical configurations. Recently, Sarkar et al. [19] endowed the literature of water waves by choosing Chebychev and Gegenbauer polynomials as basis functions to handle both half and one-third singularities.

The study of wave diffraction by the combination of thick and thin barriers using the semi-analytical method (solution of integral equations involving two types of singularities) is quite new in this field of research. This study is, first of all, about to give a promising procedure for encountering wave interaction with a pair of thin vertical partially immersed barriers between the corners of a bottom-standing thick barrier, and second, about to provide a reliable benchmark for complicated numerical solutions of integral equations having integrable singularities (half and one-third singularities). We have investigated the problem of wave interaction with a pair of partially immersed barriers in the presence of a rectangular thick bottom-standing barrier. So far as our knowledge goes these configurations of the problems have not been investigated in the literature. With the aid of Havelock’s [20] inversion formulae, the problem is reduced to a four first kind Fredholm-type integral equations involving the horizontal components of velocity as unknown functions above the corners of thick barrier and below the submerged edge of the thin barriers. The methodology of multiterm Galerkin approximation along with Chebychev and Gegenbauer polynomials has been employed to tackle half and one-third singularities near the submerged edges of the thin barriers and corners of thick barrier, respectively. To verify the present numerical model, the results of Das et al. [21] for thin vertical barriers without thick bottom-standing barrier and Mandal and Kanoria [15] for the thick bottom-standing barrier without thin vertical barriers are validated. These analyses reveal that the present simulation and the researches of Das et al. [21], Mandal and Kanoria [15] fall into a good agreement. Based on numerical results for reflection and transmission coefficients, this model shows to be useful and can be adequately brought into play as an applicable tool in marine engineering.

## Theoretical Development

*c*

_{2},

*c*

_{3}are situated at

*x*=

*s*

_{2},

*s*

_{3}, respectively, between the corners at

*x*=

*s*

_{1},

*s*

_{4}of the thick bottom-standing. The

*y*-axis is directed vertically downward, and the

*xz*-plane is considered to be the equilibrium water surface. The whole fluid region is assumed to be incompressible, nonviscous, and the flow irrotational. So that the fluid may be represented in terms of velocity potential $Re[\varphi inc(x,y)ei\tau z\u2212i\omega t]$, where $Re[]$ denotes the real part of a complex expression and

*τ*=

*k*

_{0}sin

*α*,

*κ*=

*k*

_{0}cos

*α*,

*α*is the angle of inclination and

*k*

_{0}is the unique real positive root in the regions Ω

_{1}and Ω

_{5}of the dispersion equation

*k*tanh

*kh*=

*K*,

*K*=

*ω*

^{2}/

*g*,

*ω*is the angular frequency of the incoming wave train,

*g*be the gravitational acceleration.

*τ*satisfies (cf. Kirby and Dalrymple [22]).

*α*

_{0}is the wave number for the region Ω

_{2}, Ω

_{3}, and Ω

_{4}and also the unique real positive root of the equation

*α*″tanh

*α*″

*c*=

*K*(here

*c*=

*c*

_{1}=

*c*

_{4}, gap between the free surface and the top of the thick barrier).

*ϕ*

_{j}(

*x*,

*y*) satisfies the modified Helmholtz’s equation in each subregion Ω

_{j},

*a*)

*b*)

*R*and

*T*are reflection and transmission coefficients.

*ϕ*

_{j}(

*x*,

*y*) in each of the five subdomains is given by

*a*)

*b*)

*c*)

*k*tanh

*kh*=

*K*; $\xb1i\alpha n(n=1,2,\u2026)$ are of $\alpha \u2033tanh\alpha \u2033c=K,pn=kn2+\tau 2$, $p0=k02\u2212\tau 2,qn=\alpha n2+\tau 2$, and $q0=\alpha 02\u2212\tau 2$. Here, $Rn(j)(j=1,2,3,4),Tn(j)(j=2,3,4,5)$ are unknown constants;

*V*

_{1}and

*V*

_{2}are given by $V1=Re\u22122i\kappa s1,V2=Tei\kappa (s4\u2212s1)$.

*u*

_{j}(

*y*)(

*j*= 1, 2, 3, 4) and difference of potentials

*v*

_{j}(

*y*) (

*j*= 1, 2, 3, 4) as

*u*

_{j}(

*y*) and using continuity of

*ϕ*(

*x*,

*y*) at

*x*=

*s*

_{j}, we get a system of integral equations. Equations (10) and (11) are successively used to combine those integral equations in a single one and then convert their ranges of integration into (0,

*h*). After that, the transformation (12) helps to recast the combined integral equation into the following matrix form:

**P**(

*y*,

*u*), i.e., $Pj(y,u)(j=1,2,\u202610)$ are given by

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

*i*)

*j*)

*V*

_{2}are expressed in terms of horizontal components of velocities. Thereafter, utilizing Eqs. (10)–(12), we obtain the following matrix equation:

**Q**is of the form

**Z**, the elements

*V*

_{1}and

*V*

_{2}of

**Y**are determined by solving Eq. (15). Thus, reflection and transmission coefficients

*R*and

*T*, connected to

*V*

_{1}and

*V*

_{2}, respectively, are obtained.

*N*+ 1)-term Galerkin’s approximation of

**U**(

*u*) to solve the integral equations as given by

*j*th integral equation by $\chi j(m)(u)(j=1,2,3,4)$ and integrating over (0,

*h*) successively, the following system of equation is obtained as

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

*i*)

*j*)

*k*)

*l*)

*m*)

*n*)

*o*)

*p*)

*q*)

*r*)

*s*)

*t*)

*T*

_{2n}(

*x*) the Chebychev polynomial of the first kind of order 2

*n*.

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

*i*)

*j*)

*k*)

*l*)

*m*)

*n*)

*o*)

*p*)

*q*)

*r*)

*s*)

*t*)

*u*)

*m*, respectively.

## Validation of the Numerical Model

In this section, the present numerical model is validated with the model of Das et al. [21] for two partially immersed thin vertical barrier as a special limiting case. For this purpose, the dimensionless trench height i.e., *c*_{1}/*h* (= *c*_{4}/*h*) is taken to be 0.999 so that the bathymetry of the ocean becomes almost like flat rigid bed. In Table 1, the values of |*R*| of present study are compared with the values of |*R*_{1}| and |*R*_{2}| of Das et al. [21] for *c*_{2}/*h* = 0.2 ( = *c*_{3}/*h*), *s*_{1}/*h* = 0.001, *s*_{2}/*h* = 0.3, *s*_{3}/*h* = 0.9, *s*_{4}/*h* = 2.0. It is observed from Table 1 that an agreement up to three decimal places is achieved for three different angles of incidence *α* = 0 deg, 30 deg, and 60 deg. This assures that the present breakwater model is a valid one and can be considered into various ocean engineering problems.

α = 0 deg | α = 30 deg | α = 60 deg | |||||||
---|---|---|---|---|---|---|---|---|---|

Das et al. [21] | Present result | Das et al. [21] | Present result | Das et al. [21] | Present result | ||||

Kh | $R1(lowerboundfor|R|)$ | $R2(upperboundfor|R|)$ | |R| | $R1(lowerboundfor|R|)$ | $R2(upperboundfor|R|)$ | |R| | $R1(lowerboundfor|R|)$ | $R2(upperboundfor|R|)$ | |R| |

0.2 | 0.031799 | 0.032341 | 0.032056 | 0.027843 | 0.028316 | 0.028242 | 0.016427 | 0.016705 | 0.016681 |

0.8 | 0.088445 | 0.089049 | 0.088623 | 0.081899 | 0.082476 | 0.082438 | 0.053674 | 0.053816 | 0.053799 |

1.4 | 0.126188 | 0.128896 | 0.127566 | 0.137167 | 0.139159 | 0.138231 | 0.113130 | 0.113931 | 0.113222 |

α = 0 deg | α = 30 deg | α = 60 deg | |||||||
---|---|---|---|---|---|---|---|---|---|

Das et al. [21] | Present result | Das et al. [21] | Present result | Das et al. [21] | Present result | ||||

Kh | $R1(lowerboundfor|R|)$ | $R2(upperboundfor|R|)$ | |R| | $R1(lowerboundfor|R|)$ | $R2(upperboundfor|R|)$ | |R| | $R1(lowerboundfor|R|)$ | $R2(upperboundfor|R|)$ | |R| |

0.2 | 0.031799 | 0.032341 | 0.032056 | 0.027843 | 0.028316 | 0.028242 | 0.016427 | 0.016705 | 0.016681 |

0.8 | 0.088445 | 0.089049 | 0.088623 | 0.081899 | 0.082476 | 0.082438 | 0.053674 | 0.053816 | 0.053799 |

1.4 | 0.126188 | 0.128896 | 0.127566 | 0.137167 | 0.139159 | 0.138231 | 0.113130 | 0.113931 | 0.113222 |

In order to validate the present results with the results of Mandal and Kanoria [15], we take *c*_{2}/*h* = 0.001 = *c*_{3}/*h* so that the pair of thin barriers almost vanishes. Thus, the graph of Fig. 6 in Mandal and Kanoria [15] for a thick bottom-standing barrier has been almost recovered in Fig. 2 by taking *c*_{1}/*h* = 0.5 ( = *c*_{4}/*h*), *s*_{1}/*h* = 0.001, *s*_{2}/*h* = 0.0011, *s*_{3}/*h* = 0.00111, *s*_{4}/*h* = 0.2, *α* = 45 deg. This assures that the numerical method adopted in the present article yields fairly accurate results.

In Table 2, |*R*| and |*T*| are tabulated for a pair of partially immersed thin vertical barriers just above a symmetrical trench with parameters *c*_{1}/*h* = 0.7 (= *c*_{4}/*h*), *c*_{2}/*h* = 0.2 (= *c*_{3}/*h*), *s*_{1}/*h* = 0.1, *s*_{2}/*h* = 0.3, *s*_{3}/*h* = 0.6, *s*_{4}/*h* = 0.9, *α* = 45 deg. The last column of Table 2 affirms that the energy balance relation (i.e., |*R*|^{2}+|*T*|^{2}=1) is verified. This gives the correctness of the present breakwater configuration.

Kh | |R| | |T| | |R|^{2} + |T|^{2} |
---|---|---|---|

0.25 | 0.170001 | 0.985444 | 1 |

0.50 | 0.223819 | 0.974631 | 1 |

0.75 | 0.258402 | 0.966037 | 1 |

1.00 | 0.285809 | 0.958287 | 1 |

1.25 | 0.312330 | 0.949974 | 1 |

1.50 | 0.342059 | 0.939678 | 1 |

1.75 | 0.376637 | 0.926361 | 1 |

2.00 | 0.413378 | 0.910559 | 1 |

Kh | |R| | |T| | |R|^{2} + |T|^{2} |
---|---|---|---|

0.25 | 0.170001 | 0.985444 | 1 |

0.50 | 0.223819 | 0.974631 | 1 |

0.75 | 0.258402 | 0.966037 | 1 |

1.00 | 0.285809 | 0.958287 | 1 |

1.25 | 0.312330 | 0.949974 | 1 |

1.50 | 0.342059 | 0.939678 | 1 |

1.75 | 0.376637 | 0.926361 | 1 |

2.00 | 0.413378 | 0.910559 | 1 |

Some infinite series sum appears in the numerical computations and needs to truncate the series at some finite stage without hampering the convergence of the numerical results. So, our aim is to fix the truncation size and ensure the convergence of the infinite series sums. For that reason, Table 3 is tabulated with different parametric values as *c*_{1}/*h* = 0.65 ( = *c*_{4}/*h*), *c*_{2}/*h* = 0.35 ( = *c*_{3}/*h*), *s*_{1}/*h* = 1.1, *s*_{2}/*h* = 2.2, *s*_{3}/*h* = 3.3, *s*_{4}/*h* = 4.4, *α* = 60 deg. Table 3 demonstrates that the reflection coefficients converge exactly five decimal places by fixing truncation size *r* = 100 in the infinite series sums. Thus, for the sake of convergence of the values of |*R*| and |*T*|, truncation size of infinite series sums is chosen judiciously with *r* = 100 in the subsequent numerical computations. This gives a partial check on the exactness of the present method.

Kh | r = 20 | r = 25 | r = 50 | r = 75 | r = 100 |
---|---|---|---|---|---|

0.2 | 0.347104 | 0.348225 | 0.348573 | 0.348582 | 0.348584 |

0.4 | 0.398555 | 0.399789 | 0.399901 | 0.399945 | 0.399947 |

0.6 | 0.397169 | 0.398367 | 0.398666 | 0.398677 | 0.398679 |

0.8 | 0.367029 | 0.367945 | 0.368091 | 0.368115 | 0.368118 |

1.0 | 0.313526 | 0.314085 | 0.314255 | 0.314264 | 0.314265 |

Kh | r = 20 | r = 25 | r = 50 | r = 75 | r = 100 |
---|---|---|---|---|---|

0.2 | 0.347104 | 0.348225 | 0.348573 | 0.348582 | 0.348584 |

0.4 | 0.398555 | 0.399789 | 0.399901 | 0.399945 | 0.399947 |

0.6 | 0.397169 | 0.398367 | 0.398666 | 0.398677 | 0.398679 |

0.8 | 0.367029 | 0.367945 | 0.368091 | 0.368115 | 0.368118 |

1.0 | 0.313526 | 0.314085 | 0.314255 | 0.314264 | 0.314265 |

## Numerical Results and Analysis

In this section, a computer program in Mathematica software has been developed based on the theory discussed in the aforementioned sections and numerical simulations are performed with the different nondimensional parametric values. To understand the effect of various physical parameters like angle of incidence, height of thick barrier, length of thin barrier, width of thick barrier, etc., on wave transformation due to thin and thick barriers, few results of reflection and transmission coefficients are graphically explored. These numerical estimates of |*R*| and |*T*| will provide global information on the amount of wave energy reflected back by thick and thin barriers and transmitted on the lee side of the thick barrier which gives a quick assessment on wave transformation by the thick and thin barriers.

Figure 3 presents the effect of the angle of incidence on the wave reflection and transmission. Four graphs are plotted for |*R*| and |*T*| against wave number with four different angles of incidence *α* = 0 deg, 30 deg, 45 deg, and 60 deg. The other dimensionless parameters are taken as *c*_{1}/*h* = 0.65 (= *c*_{4}/*h*), *c*_{2}/*h* = 0.15 (= *c*_{3}/*h*), *s*_{1}/*h* = 0.25, *s*_{2}/*h* = 0.5, *s*_{3}/*h* = 0.75, *s*_{4}/*h* = 1.0. Figure 3 shows that more waves are reflected and less waves are transmitted with a decrease in the angle of incidence.

The influence of the height of the thick barrier is revealed in Fig. 4. Under the consideration of the parametric values *c*_{2}/*h* = 0.3 (= *c*_{3}/*h*), *s*_{1}/*h* = 1.0, *s*_{2}/*h* = 2.0, *s*_{3}/*h* = 3.0, *s*_{4}/*h* = 4.0, *α* = 45 deg, reflection and transmission coefficients are depicted against *Kh* in Figs. 4(a) and 4(b). It is observed from these figures that the global maxima of |*R*| increase and the global minima of |*T*| decrease as the height of the thick bottom-standing barrier increases. It is also noted from Fig. 4(a) that peak values of |*R*| have the tendency to shift to the right as the height of the thick barrier increases.

In Fig. 5, |*R*| and |*T*| are plotted against *Kh* with various nondimensional parameters *c*_{1}/*h* = 0.6 ( = *c*_{4}/*h*), *s*_{1}/*h* = 0.6, *s*_{2}/*h* = 1.2, *s*_{3}/*h* = 1.4, *s*_{4}/*h* = 1.8, *α* = 60° to analyze the impact of length of the pair of thin barriers. Figure 5(a) exhibits that the amount of reflection increases with the length of barriers. On the other hand, Fig. 5(b) manifests totally contrasting nature for the transmission coefficient. This phenomenon occurs due to larger barrier length helps more waves to reflect and less to transmit.

Curves for absolute estimation of |*R*| and |*T*| are shown in Fig. 6 for different separation lengths of the thin barriers. A set of the dimensionless parameters is chosen as *c*_{1}/*h* = 0.65 ( = *c*_{4}/*h*), *c*_{2}/*h* = 0.35 ( = *c*_{3}/*h*), *s*_{1}/*h* = 0.25, *s*_{4}/*h* = 3.25, *α* = 30°. From Fig. 6(a), it is noteworthy to mention that the spikes of |*R*| increase as the barrier spacing of thin barriers increases. Figure 6(b) infers that minima of the spikes of |*T*| decrease when the separation lengths of the thin barrier increase.

The width of the thick bottom-standing barrier has made a severe impact on reflection and transmission coefficients as shown in Fig. 7. |*R*| and |*T*| are both graphically visualized against wave number *Kh* for three different barrier widths. For this graphical depiction, the nondimensional parameters are considered as *c*_{1}/*h* = 0.8 ( = *c*_{4}/*h*), *c*_{2}/*h* = 0.4 ( = *c*_{3}/*h*), *s*_{2}/*h* = 2.0, *s*_{3}/*h* = 2.5, *α* = 60 deg. From both Figs. 7(a) and 7(b), it is concluded that some spikes are observed at some discrete frequencies for both |*R*| and |*T*|. The maxima of the spikes of |*R*| and minima of the spikes of |*T*| have the tendency to shift to the left as the width of the thick barrier increases. Moreover, an interesting fact has been discovered in Figs. 4, 6, and 7. The magnitudes of reflection and transmission coefficients have sudden spikes at some certain frequencies that may be termed as “resonant frequencies.” This is mainly due to the resonant effects in the presence of multiple obstacles in the form of two thin partially immersed vertical barriers and a bottom-standing rectangular thick barrier in a same-barrier configuration.

## Conclusion

This paper presents a mathematical treatment to put a light on wave diffraction by pair of thin vertical partially immersed barriers just above a thick bottom-standing barrier. The prime objective of this study is to address both one-third and half singularities that arise in the integral equations and to provide a promising procedure for handling such complicated numerical calculations. The analytical expressions giving reflection and transmission coefficients are shown graphically against wave number to analyze the effects of various physical parameters. The present results are reproduced and validated with the known results of Das et al. [21] and Mandal and Kanoria [15] in the literature to verify the accuracy of this methodology. From the analysis of illustrations, it is pointed out that the angle of incidence has a crucial impact on the magnitude of the reflection and transmission coefficients. It is observed from the graphical depictions that the global maxima of |*R*| and global minima of |*T*| are varying with the height thick bottom-standing barrier. When the height of bottom-standing barrier increases, then peaks of |*R*| have a propensity to shift to the right. Increasing the length of thin barriers produces more reflection consequently, less transmission to the seaside, yielding less impact on the seashore by a pair of thin vertical partially immersed barriers if imposed at the free surface. Both separation lengths of thin barriers and width of thick barrier have made a severe impact on maxima of peaks of |*R*| and minima of peaks of |*T*|. From all these curves, one can predict the wave numbers in which maximum reflection and minimum transmission occur. Some resonant frequencies are also observed in some figures due to multiple obstacles in the form of thin and thick barriers.

This study will be useful to solve integral equations having one-third as well as half singularities which arise in the other areas of mathematical physics. This investigation will be of immense help in the prediction of offshore inundation during extreme events like storm surges and tsunamis. Finally, the authors believe that the present work may provide some interesting information for experimental scientists/marine researchers/seismologists working on this relevant topic.

## Acknowledgment

The authors are very much indebted to the reviewers for their valuable suggestions which enabled the authors to carry out the desired revision of the manuscript. This work is partially supported by a SERB, DST(TAR/2022/000107).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.